Heightened Commitment

According to halacha it is sufficient to light one candle each night
throughout Chanukah for the entire household. Those who are mehadrin
min hamehadrin extremely careful in performing mitzvot follow Beit
Hillel and add one candle each night. (Shabbat 21b)

QUESTION:

Why, in regard to lighting the Menorah, all homes conduct
themselves in the manner of mehaderin min hamehaderin
supremely scrupulous while in many other mitzvot they follow
lesser halachic requirements?

ANSWER:

In describing the miracle of Chanukah, the Gemara relates that
the Jews found only one cruse of oil, and that it had the seal
of the Kohen Gadol. In the Beit Hamikdash there were Kohanim
assigned to the special task of oil making. It was not the
responsibility of the Kohen Gadol to make oil. Why then did
this particular cruse bear the Kohen Gadol's seal?

The Kohen Gadol was required to bring a daily sacrifice
consisting of flour and oil, known as "chavitei Kohen Gadol"
(Vayikra 6:15). Normally, the oil used for this offering
would be of lower quality than that used for the kindling of
the Menorah. However, the Kohen Gadol in that time was a
highly distinguished spiritual personality, and a mehader
bemitzvot scrupulous in mitzvot who used pure olive oil for
his daily sacrifice.

When the Hasmoneans entered the Beit Hamikdash, they did not
find any oil to kindle the Menorah. Luckily they found one
cruse which was for the Kohen Gadol's daily sacrifice, and to
their amazement it was pure olive oil. Were it not for the
fact that this Kohen Gadol was a mehader bemitzvot, no oil
would have been available. To emphasize the uniqueness
of the Kohen Gadol at that time, we emulate his actions in the
form of mehaderin min hamehaderin.

On the basis of the above, we find another rationale for the
observance of Chanukah for eight days, though there actually
was oil for the first night.

The Menorah required one half lug for each of the seven
candles, adding up to a total of 3 1/2 lugim (Menachot 88a).
The Kohen Gadol's daily sacrifice required only a total of
three lugim of oil per day (Menachot 87b). Thus, the cruse
found was not sufficient for even one night, though
miraculously it burnt through the entire night.